2024/2025
Введение в теорию случайных процессов
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Бланк Михаил Львович
Язык:
английский
Кредиты:
3
Course Syllabus
Abstract
The course is a continuation of the standard course in probability theory (associated mainly with combinatorics) and is intended for an initial introduction to the theory of random processes. Special attention is paid to the connection of this theory with functional analysis and the general measure theory. The course is aimed at bachelors 2–4 courses, undergraduates and graduate students.
Learning Objectives
- This course is aimed at providing students with a solid working knowledge in the basic concepts, important techniques and examples in theory of random processes. Learn probabilistic methods for analyzing random processes and proving mathematical theorems. Develop probabilistic intuition.
Expected Learning Outcomes
- Student is expected to be able to construct probability measures on infinite dimensional spaces and, in particular, function spaces
- Student would learn to apply stochastic calculus to derive solutions of stochastic differential equations and to study their properties
- The student is expected be able to use measure-theoretic and analytic techniques for the derivation of equations describing Markov and diffusion processes
- The students are expected to be able to outline proofs of important theorems of continuous-time martingale processes
Course Contents
- The concept of a random process.
- Elements of random analysis.
- Correlation theory of random processes.
- Markov processes with discrete and continuous time
- Wiener and Poisson processes.
- Stochastic integral. Ito’s formula.
- (sub/super) martingales.
- Infinitesimal semigroup operator.
- Stochastic stability of dynamical systems.
- Large deviations in Markov processes and chaotic dynamics.
- Nonlinear Markov processes.
Bibliography
Recommended Core Bibliography
- Grimmett, G., & Welsh, D. J. A. (2014). Probability : An Introduction (Vol. 2nd ed). Oxford: OUP Oxford. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=852090
- Krylov, N. V. (2002). Introduction to the Theory of Random Processes. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971029
Recommended Additional Bibliography
- Chaumont, L., & Yor, M. (2012). Exercises in Probability : A Guided Tour From Measure Theory to Random Processes, Via Conditioning (Vol. 2nd ed). Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=466664
- Stirzaker, D. (2003). Elementary Probability (Vol. 2nd ed). Cambridge, UK: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=125155